| 释义 |
quadratic reciprocity Given two distinct, odd primes p and q, then p is a quadratic residue modulo q if and only if q is a quadratic residue modulo p, unless p ≡ q ≡ 3 (mod 4), in which case precisely one of p,q is a quadratic residue modulo the other. This was proved by Gauss around 1801 but had been suspected since Euler's time. As an application, note that it is not immediately obvious whether x2 ≡ 31 (mod 257) has a solution, but now note, in terms of the Legendre symbol, that (31|257) = (257|31) = (9|31) = 1 as 9 is clearly a quadratic residue and so 31 is a quadratic residue modulo 257. The theorem does not provide guidance on what the two square roots are—in this case ±51.
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